3.2PureStretch

Quadratic Simultaneous Equations

When one equation is linear and the other is quadratic, substitution is the way in. You end up with a single quadratic to solve, giving up to two solution pairs — the points where a line meets a curve.

30 min Video by Zeeshan Zamurred Equations and Inequalities

Edexcel AS level Maths: 3.2 Quadratic Simultaneous EquationsWatch the full walkthrough before the notes below.

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What you'll be able to do

Solve a linear–quadratic pair by substitution
Obtain and solve the resulting quadratic
Find both coordinates of each solution
Link the number of solutions to line–curve intersections

Always substitute the linear equation

Rearrange the $linear$ equation to make one variable the subject, then substitute it into the quadratic. This produces a single quadratic in one variable.

1Substitute

y = x + 1

x^{2} + (x + 1)^{2} = 25

2Expand:

x^{2} + x^{2} + 2 x + 1 = 25 \Rightarrow 2 x^{2} + 2 x - 24 = 0

3Divide by 2 and factorise:

x^{2} + x - 12 = (x + 4) (x - 3) = 0

4So

x = - 4

x = 3

; then

y = x + 1

gives

y = - 3

y = 4

Answer

(- 4, - 3)

and

(3, 4)

Tip — Substitute the linear into the quadratic — never the other way round.

Pairing up the answers

Each $x$ -value has its own matching $y$ -value. Use the linear equation (the simplest one) to find each $y$ , and keep the pairs together.

Tip — Two x-values give two coordinate pairs. Do not mix them up.

Line meets curve

These solutions are the intersection points of a line and a curve. Two solutions means the line is a chord; one (repeated) means it is a tangent; none means it misses the curve entirely — exactly the discriminant idea again.

Formula recap

y = m x + c into y = a x^{2} + b x + c

Substitute linear into quadratic.

2 / 1 / 0 solutions

Chord / tangent / miss.

Common mistakes to avoid

Pairing every x with every y, giving four “solutions”.

Each x has one matching y from the linear equation — keep the pairs linked.

Substituting the quadratic into the linear.

Always substitute the linear equation into the quadratic.

Key takeaways

Make a variable the subject of the linear equation and substitute into the quadratic.
Solve the resulting single quadratic.
Find the matching second coordinate for each solution.
Solutions are line–curve intersections (chord/tangent/miss).

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